A magnetic multi-layer soft robot for on-demand targeted adhesion

Magnetic soft robots have shown great potential for biomedical applications due to their high shape reconfigurability, motion agility, and multi-functionality in physiological environments. Magnetic soft robots with multi-layer structures can enhance the loading capacity and function complexity for targeted delivery. However, the interactions between soft entities have yet to be fully investigated, and thus the assembly of magnetic soft robots with on-demand motion modes from multiple film-like layers is still challenging. Herein, we model and tailor the magnetic interaction between soft film-like layers with distinct in-plane structures, and then realize multi-layer soft robots that are capable of performing agile motions and targeted adhesion. Each layer of the robot consists of a soft magnetic substrate and an adhesive film. The mechanical properties and adhesion performance of the adhesive films are systematically characterized. The robot is capable of performing two locomotion modes, i.e., translational motion and tumbling motion, and also the on-demand separation with one side layer adhered to tissues. Simulation results are presented, which have a good qualitative agreement with the experimental results. The feasibility of using the robot to perform multi-target adhesion in a stomach is validated in both ex-vivo and in-vivo experiments.


Calculation of magnetic field
We establish a cylindrical coordinate system with the center of the magnet as the origin O (Fig. 4a).In the translational motion of the robot, the magnetic field can be expressed as B e e , where e r and e z are the positive unit vector of the x -axis and z -axis, respectively.x B and z B can be written as 1 : ) Here, ( , ) r z denotes the coordinates of the center point of the robot.0  is the permeability of the vacuum.M is the scalar magnetization of the cylindrical permanent magnet.L and a are the thickness and radius of the magnet, respectively.

Calculation of the magnetic force
Additionally, we assume that the magnetization profile m perpendicular to its surface is uniformly distributed shown in Supplementary Fig. 1c.Therefore, the magnetic force exerted on the robot can be expressed as: where m V represents the volume of the magnetic portion, and  can be expressed as e e x z x z . The decomposition of F m into the x -axis and z -axis can be expressed as and z m F are both positive because the magnetic force is attractive between the magnet and the robot (Supplementary Fig. 3).

Calculation of the friction force
The friction force acting on the robot can be expressed as , given the normal force acting on it as N F , where Analyzing the tumbling motion of the robot by setting the desired magnetic field B .Firstly, the initial direction of the magnetic field is the positive y -axis direction (Fig. 4f and Supplementary Fig. 7).When B starts to rotate clockwise, the magnetic moment m tends to align with the direction of B .The robot then achieves a tumbling motion actuated by the magnetic torque.
The magnetic field B can be expressed as: where f represents the rotation frequency of the magnetic field.
The magnetic moment m can be expressed as: The magnetic torque m τ can be expressed as: where R is the standard z -axis rotational matrix that accounts for the change in the direction of m due to the robot's flip: where   φ t is the angular displacement of the robot.Substituting Eq. ( 5), (6), and (7) into Eq.( 4), m τ can be written as: As the robot tumbles, a supporting force F is provided at the contact surface (Supplementary Fig. 7).
Assuming the centroid of the robot undergoes a parabolic motion, the direction of F is parallel to the motion direction of its centroid.Therefore, F can be expressed as: where f v represents the velocity of the centroid of the robot.Assuming that the robot does not slip as it rolls on the gastric tissue, the kinematic relationship between the f v and   φ t can be expressed as: Treating the tumbling motion as rotation around a fixed point, the conservation of momentum principle can be used to obtain: The above equation can be rewritten as: where f C , M , l , and J represent tumbling damping, the mass, length, and moment of inertia of the robot, respectively.Assuming that the robot does not slip during the tumbling process and can quickly reach a steady-state velocity under the influence of a magnetic field 3,4 , Eq. ( 13) can be rewritten as: When the robot can achieve stable tumbling motion, the angle between the driving field and the angular It can be observed that once the rotation frequency of the magnetic field is higher than the step-out frequency of the robot, i.e., 2π step-out f φ   , it becomes difficult to predict the angular velocity of the robot.
At the same time, according to Eq. (15), it is found that in a stable tumbling motion, as the field strength increases, the step-out frequency of the robot increases.